MACS 30100
University of Chicago
\[Y = \beta_0 + \beta_{1}X_1\]
Arithmetic mean
\[\bar{x} = \frac{1}{n} \sum_{i = 1}^n x_i\]
Variance
\[E[X] = \mu\]
\[\text{Var}(X) \equiv \sigma^2 = E[X^2] - (E[X])^2\]
Standard deviation
\[\sigma = \sqrt{E[X^2] - (E[X])^2}\]
Median absolute deviation
\[MAD = \text{median}(|X_i - \text{median}(X)|)\]
Nonparametric density estimation
\[x_0 + 2(j - 1)h \leq X_i < x_0 + 2jh\]
\[\hat{p}(x) = \frac{\#_{i = 1}^n [x_0 + 2(j - 1)h \leq X_i < x_0 + 2jh]}{2nh}\]
\[\hat{p}(x) = \frac{\#_{i = 1}^n [x_0 + 2(j - 1)h \leq X_i < x_0 + 2jh]}{2nh}\]
\[\hat{p}(x) = \frac{1}{nh} \sum_{i = 1}^n W \left( \frac{x - X_i}{h} \right)\]
\[W(z) = \begin{cases} \frac{1}{2} & \text{for } |z| < 1 \\ 0 & \text{otherwise} \\ \end{cases}\]
\[z = \frac{x - X_i}{h}\]
Kernels
\[\hat{x}(x) = \frac{1}{nh} \sum_{i = 1}^k K \left( \frac{x - X_i}{h} \right)\]
\[K(z) = \frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2} z^2}\]
\[K(z) = \frac{1}{2} \mathbf{1}_{\{ |z| \leq 1 \} }\]
\[K(z) = (1 - |z|) \mathbf{1}_{\{ |z| \leq 1 \} }\]
\[K(z) = \frac{15}{16} (1 - z^2)^2 \mathbf{1}_{\{ |z| \leq 1 \} }\]
\[K(z) = \frac{3}{4} (1 - z^2) \mathbf{1}_{\{ |z| \leq 1 \} }\]
\[h = 0.9 \sigma n^{-1 / 5}\]
\[A = \min \left( S, \frac{IQR}{1.349} \right)\]
\[\mu = E(\text{Income}|\text{Education}) = f(\text{Education})\]
\[\mu = E(Y|x) = f(x)\]
\[X_1 \in \{1, 2, \dots ,10 \}\] \[X_2 \in \{1, 2, \dots ,10 \}\] \[X_3 \in \{1, 2, \dots ,10 \}\]
\[\mu = E(Y|x_1, x_2, x_3) = f(x_1, x_2, x_3)\]
\[\hat{f}(x_0) = \frac{1}{K} \sum_{x_i \in N_0} y_i\]
\[f(x) = 2 + x + \epsilon_i\]
\[f(x) = 2 + x + x^2 + x^3 + \epsilon_i\]
\[f(x) = 2 + x + x^2 + x^3 + \epsilon_i\]
\[\text{Distance}(x_i, y_i) = \left( \sum_{i = 1}^n |x_i - y_i| ^p \right)^\frac{1}{p}\]
\[\Pr(Y = j | X = x_0)\]
\[1 - E \left( \max_j \Pr(Y = j | X) \right)\]
\(Pr(Y = j| X = x_0) = \frac{1}{K} \sum_{i \in N_0} I(y_i = j)\)$